Beatty sequences

Suppose R is an irrational
number greater than 1, and let S be the number satisfying the
equation 1/R + 1/S = 1. Let [x] denote the floor
function of x, that is, the greatest integer less than or
equal to x. Then the sequences [nR] and [nS],
where n ranges through the set N of positive integers,
are the Beatty sequences determined by R. The interesting thing
about them is that they partition N; in other words, every positive
integer occurs exactly once in one sequence or the other. For example, when
R is the golden ratio (about
1.618), the two sequences begin with

1, 3, 4, 6, 8, 9, 11, 12, 14, 16, 17, 19, 21, ..., and

2, 5, 7, 10, 13, 15, 18, 20, 23, 26, 28, 31, 34....

Beatty sequences are named after the American mathematician Samuel Beatty
(1881–1970) who introduced them in 1926 in a problem in the American
Mathematical Monthly. Beatty was the first person to receive a Ph.D.
in mathematics from a Canadian university, a colorful teacher, and a problemist
who became the chairman of the mathematics department, and later, Chancellor,
of the University of Toronto.